Optimal. Leaf size=95 \[ -\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}-\frac {4 b \sin (e+f x)}{15 f (b \sec (e+f x))^{3/2}}+\frac {8 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}} \]
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Rubi [A] time = 0.10, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2627, 3771, 2639} \[ -\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}-\frac {4 b \sin (e+f x)}{15 f (b \sec (e+f x))^{3/2}}+\frac {8 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2627
Rule 2639
Rule 3771
Rubi steps
\begin {align*} \int \frac {\sin ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx &=-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac {2}{3} \int \frac {\sin ^2(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx\\ &=-\frac {4 b \sin (e+f x)}{15 f (b \sec (e+f x))^{3/2}}-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac {4}{15} \int \frac {1}{\sqrt {b \sec (e+f x)}} \, dx\\ &=-\frac {4 b \sin (e+f x)}{15 f (b \sec (e+f x))^{3/2}}-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac {4 \int \sqrt {\cos (e+f x)} \, dx}{15 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\\ &=\frac {8 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {4 b \sin (e+f x)}{15 f (b \sec (e+f x))^{3/2}}-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 63, normalized size = 0.66 \[ \frac {-68 \sin (2 (e+f x))+10 \sin (4 (e+f x))+\frac {192 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{\sqrt {\cos (e+f x)}}}{360 f \sqrt {b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {b \sec \left (f x + e\right )}}{b \sec \left (f x + e\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (f x + e\right )^{4}}{\sqrt {b \sec \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.20, size = 328, normalized size = 3.45 \[ -\frac {2 \left (5 \left (\cos ^{6}\left (f x +e \right )\right )+12 i \cos \left (f x +e \right ) \sin \left (f x +e \right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-12 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )+12 i \sin \left (f x +e \right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-12 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )-16 \left (\cos ^{4}\left (f x +e \right )\right )+23 \left (\cos ^{2}\left (f x +e \right )\right )-12 \cos \left (f x +e \right )\right ) \sqrt {\frac {b}{\cos \left (f x +e \right )}}}{45 f \sin \left (f x +e \right ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (f x + e\right )^{4}}{\sqrt {b \sec \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (e+f\,x\right )}^4}{\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{4}{\left (e + f x \right )}}{\sqrt {b \sec {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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